初中毕业学业考试试卷
数 学
考生注意:本学科试卷共三道大题25小题,满分120分,考试时量120分钟. 一、选择题(本大题共8小题,每小题3分,满分24分) 1.在实数0
1
3
-,0.74,π中,无理数有( ) A .1个
B .2个
C .3个
D .4个
2.用计算器求32值时,需相继按“2”,“∧”,“3”,“=
,“2”,
“∧”,“4”,“=”键,则输出结果是( )
A .4
B .5
C .6
D .16 3.下图所示的几何体的主视图是( ) 4.不等式组475(1)22463x x x x -<-??->-?
的解集在数轴上表示为( )
5.下列运算正确的是( ) A .2
2
21ab ab -= B .tan 45sin 451=g °° C .2
3x x x =g
D .23
5
()a a =
6.下列不是必然事件的是( )
A .两直线相交,对顶角相等
B .三角形的外心到三个顶点的距离相等
C .三角形任意两边之和大于第三边
D .两相似多边形面积的比等于周长的比
7.如图,AB CD ∥,且1115∠=°,75A ∠=°, 则E ∠的度数是( ) A .30° B .50°
C .40°
D .60°
8.为了预防“HINI ”流感,某校对教室进行药熏消毒,药品燃烧时,室内每立方米的含药量与时间成正比;燃烧后,室内每立方米含药量与时间成反比,则消毒过程中室内每立方米含药量y 与时间t 的函数关系图象大致为( )
A .
B .
C .
D .
A .
B .
C .
D .
E
D C A B 1
二、填空题(本大题共8小题,每小题3分,满分24分) 9
.的绝对值为 .
10.如图,O e 是ABC △的内切圆,与边BC CA AB ,,
的切点分别为D E F ,,,若70A ∠=°,则EDF ∠= . 11.张家界国际乡村音乐周活动中,来自中、日、美的三名音乐家准备在同一节目中依次演奏本国的民族音乐,若他们出场先后的机会是均等的,则按“美—日—中”顺序演奏的概率是 .
12.将函数33y x =-+的图象向上平移2个单位,得到函数 的图象. 13.分解因式32
a a
b -= .
14.我市甲、乙两景点今年5月上旬每天接待游客的人数如图所示,甲、乙两景点日接待游
客人数的方差大小关系为:2S 甲 2
S 乙.
15.对于正实数a b ,
作新定义:a b a b *=+,在此定义下,若955x *=,则x 的值为 .
16.如图,等腰梯形ABCD 中,AD BC ∥,且1
2
AD BC =
,E 为AD 上一点,AC 与BE 交于点F ,若:2:1AE DE =,则
AEF CBF 的面积
的面积
△△= . 三、解答题(本题共9小题,满分72分) 17.(本小题6分)
计算1
1(52sin 452-??
++-+ ???
°°
A .
B .
C .
D .
A
E D
C
F
B
18.(本小题6分)
小明将一幅三角板如图所示摆放在一起,发现只要知道其中一边的长就可以求出其它各边的长,若已知2CD =,求AC 的长.
19.先化简,后求值(本小题6分)
2
421
422
a a a +--+-
其中2a = 20.(本小题6分)
在建立平面直角坐标系的方格纸中,每个小方格都是边长为1的小正方形,ABC △的顶点均在格点上,点P 的坐标为(10)-,,请按要求画图与作答
(1) 把ABC △绕点P 旋转180°得A B C '''△. (2)把ABC △向右平移7个单位得A B C ''''''△.
(3)A B C '''△与A B C ''''''△是否成中心对称,若是,找出对称中心P ',并写出其坐标.
21.列方程解应用题(本小题9分)
“阳黄公路”开通后,从长沙到武陵源增加了一条新线路,新线路里程在原线路长360Km 的基础上缩短了50Km ,今有一旅游客车和小车同时从长沙出发前往武陵源,旅游客车走新线路,小车因故走原线路,中途停留6分钟.若小车速度是旅游客车速度的1.2倍,且两车同时到达武陵源,求两车的速度各是多少?
D B A
C
如图,有两个动点E F ,分别从正方形ABCD 的两个顶点B C ,同时出发,以相同速度分别沿边BC 和CD 移动,问:
(1)在E F ,移动过程中,AE 与BF 的位置和大小有何关系?并给予证明. (2)若AE 和BF 相交点O ,图中有多少对相似三角形?请把它们写出来.
23.(本小题9分)
我市今年初三体育考试结束后,从某县3000名参考学生中抽取了100名考生成绩进行统计分析(满分100分,记分均为整数),得到如图所示的频数分布直方图,请你根据图形完成下列问题:
(1)本次抽样的样本容量是 . (2)请补全频数分布直方图.
(3)若80分以上(含80分)为优秀,请你据此.估算该县本次考试的优秀人数.
24.(本小题9分) 有若干个数,第1个数记为1a ,第2个数记为2a ,第3个数记为3a ,L 第n 个数记为n a ,若11
3
a =-
,从第二个数起,每个数都等于.............1.与前面那个数的差的倒数............ (1)分别求出234a a a ,,的值. (2)计算12336a a a a ++++L 的值.
F D C E
O B
A
分数
在平面直角坐标系中,已知(40)A -,,(10)B ,,且以AB 为直径的圆交y 轴的正半轴于点
(02)C ,,过点C 作圆的切线交x 轴于点D .
(1)求过A B C ,,三点的抛物线的解析式 (2)求点D 的坐标
(3)设平行于x 轴的直线交抛物线于E F ,两点,问:是否存在以线段EF 为直径的圆,恰好与x
初中毕业学业考试数学试卷答案
一、选择题 1.B 2.A 3.B 4.A 5.C 6.D 7.C 8.A
二、填空题
9
10.55°
11.
16
12.35y x =-+
13.()()a a b a b +- 14.22
S S >乙甲
15.16
16.
19
三、解答题
17.原式212
2=+-?
························································· 3分
211)=+ ···················································································· 4分
211=+ ······················································································ 5分
2= ·
············································································································ 6分 18.解:2BD CD ==Q
BC ∴==················································································ 2分
AB x ∴=,则2AC x =
222(2)x x ∴+= ···················································································· 4分
3
x ∴=
··································································································· 5分
2AC AB ==
························································································ 6分 19.解:原式421
(2)(2)22
a a a a =
+-+-+-
42(2)2
(2)(2)(2)(2)(2)(2)
a a a a a a a a -+=
+-+-+-+- ················································· 2分
42(2)(2)
(2)(2)
a a a a +--+=
+- ··················································································· 3分
12
a =
+ ········································································································ 4分
当2a =
时
1
1a =+
1
== ··························································································6分20.注:每问2分
(3)(2.50)
P',
21.解:设旅游客车速度为x Km/h,则小车为1.2x Km/h ········································1分3601310
1.210
x x
+=····························································································3分解方程得100
x= ···························································································7分经检验120
x=是方程的根,且合题意1.2100120
?=Km/时 ···································8分答:小车的平均速度为120Km/时·······································································9分22.解:(1)在正方形ABCD中,AB BC
=,90
ABC BCD
∠=∠=°
BE CF
=
Q ··································································································1分ABE BCF
∴△≌△(SAS)············································································2分EAB FBC
∴∠=∠ ·························································································3分90
CBF ABO
∠+∠= Q°·················································································4分90
EAB ABO
∴∠+∠=°
在ABO
△中,180()90
AOB EAB ABO
∠=-∠+∠=
°°
AE BF
∴⊥··································································································6分(2)有5对相似三角形 ···················································································7分ABO BEO
△∽△ABO AEB
△∽△BEO BFO
△∽△
ABE BCF
△∽△ABO BFC
△∽△··························································9分
23.(1)100 ··································································································2分(2) ···········································································································5分(3)30000.61800
?=
该县优秀人数约为1800人 ················································································9分
分数
24.解:(1)2113
41413
3a =
==??
-- ??? ································································· 2分 31
1
431144a =
=
=-
·
························································································ 4分 411
143a ==-- ·
···························································································· 6分 (2)123361
3412533
4a a a a ??+++=-
++?= ???L ··············································· 9分
25.解:(1)令二次函数2
y ax bx c =++,则
1640
2a b c a b c c -+=??
++=??=?
··························································································· 1分 12322a b c ?=-??
?
∴=-??=???
··································································································· 2分 ∴过A B C ,,三点的抛物线的解析式为213
222
y x x =--+ ·································· 4分
(2)以AB 为直径的圆圆心坐标为3
02
O ??' ???
,
52O C '∴=
32
O O '= ···················································································· 5分 CD Q 为圆O '切线 OC CD '∴⊥·
···································································· 6分 90O CD DCO '∴∠+∠=°
90CO O O CO ''∠+∠=° CO O DCO '∴∠=∠
O CO CDO '∴△∽△ //O O OC OC OD '=
······················································ 8分 3/22/2OD = 83OD ∴= D ∴坐标为803??
???
, ························································································· 9分 (3)存在 ··································································································· 10分 抛物线对称轴为3
2
X =-
设满足条件的圆的半径为r ,则E 的坐标为3()2r r -
+,或3()2
F r r --, 而E 点在抛物线213
222
y x x =-
-+上 21333
()()22222
r r r ∴=--+--++
112r ∴=-+
212
r =--
故在以EF 为直径的圆,恰好与x 轴相切,该圆的半径为1-+,1+ ········· 12分 注:解答题只要方法合理均可酌情给分